Berry-Esseen bounds of second moment estimators for Gaussian processes observed at high frequency
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Let Z:={Z_t,t≥0} be a stationary Gaussian process. We study two estimators of 𝔼[Z_0^2], namely f_T(Z):= 1/T∫_0^T Z_t^2dt, and f_n(Z) :=1/n∑_i =1^n Z_t_i^2, where t_i = i Δ_n, i=0,1,…, n, Δ_n→ 0 and T_n := n Δ_n→∞. We prove that the two estimators are strongly consistent and establish Berry-Esseen bounds for a central limit theorem involving f_T(Z) and f_n(Z). We apply these results to asymptotically stationary Gaussian processes and estimate the drift parameter for Gaussian Ornstein-Uhlenbeck processes.
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